School on Moduli Spaces
Wednesday 5th January 2011 to Friday 14th January 2011
09:00 to 09:55  Registration  
09:55 to 10:00  Welcome from Sir David Wallace (INI Director)  
10:00 to 11:00 
Augmented bundles  I
The notion of augmented bundle comprises various extra structures on vector bundles. The lectures deal with examples like quasiparabolic bundles, coherent systems, framed bundles, Higgs bundles, and holomorphic triples, all on a compact Riemann surface. I'll explain their coarse moduli spaces and discuss the role of the stability parameters.

INI 1  
11:00 to 11:30  Morning coffee  
11:30 to 12:30 
T Gómez (Universidad Complutense de Madrid) Principal bundles  I
This is an introduction to principal bundles in algebraic geometry. We will start with the basic definitions (associated fibrations, reduction and extension of structure group, stability, etc...) and then give a sketch of the construction of the moduli
Notes: The following are the notes of some lectures I gave in Warsaw. In that school I gave 5 lectures, so this time I won't be able to cover all this material, but this could serve as a reference:
T. Gomez, Lectures on principal bundles over projective varieties, Pragacz, P. (Ed.) Algebraic cycles, sheaves, shtukas, and moduli. Impanga lecture notes. Trends in Mathematics. Birkh\"auser
Verlag, Basel, 2008. viii+236 pp., 4568. Lectures on Principal Bundles over proyective varieties. arXiv: 1011.1801v1

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Geometric Invariant Theory  I  INI 1  
15:00 to 15:30  Afternoon tea  
15:30 to 16:30 
N Nitsure (Tata Institute of Fundamental Research) Deformation Theory : an introduction to the algebraic approach  I
This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples.
The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangentobstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces.

INI 1  
16:45 to 17:45  Tutorials  INI 1  
17:45 to 18:45  Welcome wine reception  
18:45 to 19:30  Dinner at Wolfson Court (residents only) 
10:00 to 11:00 
Augmented bundles  II
The notion of augmented bundle comprises various extra structures on vector bundles. The lectures deal with examples like quasiparabolic bundles, coherent systems, framed bundles, Higgs bundles, and holomorphic triples, all on a compact Riemann surface. I'll explain their coarse moduli spaces and discuss the role of the stability parameters.

INI 1  
11:00 to 11:30  Morning coffee  
11:30 to 12:30 
T Gómez (Universidad Complutense de Madrid) Principal bundles  II
This is an introduction to principal bundles in algebraic geometry. We will start with the basic definitions (associated fibrations, reduction and extension of structure group, stability, etc...) and then give a sketch of the construction of the moduli
Notes: The following are the notes of some lectures I gave in Warsaw. In that school I gave 5 lectures, so this time I won't be able to cover all this material, but this could serve as a reference:
T. Gomez, Lectures on principal bundles over projective varieties, Pragacz, P. (Ed.) Algebraic cycles, sheaves, shtukas, and moduli. Impanga lecture notes. Trends in Mathematics. Birkh\"auser
Verlag, Basel, 2008. viii+236 pp., 4568. Lectures on Principal Bundles over proyective varieties. arXiv: 1011.1801v1

INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Geometric Invariant Theory  II  INI 1  
15:00 to 15:30  Afternoon tea  
15:30 to 16:30 
N Nitsure (Tata Institute of Fundamental Research) Deformation Theory : an introduction to the algebraic approach  II
This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples.
The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangentobstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces.

INI 1  
16:45 to 17:45  Tutorials  INI 1  
18:45 to 19:30  Dinner at Wolfson Court (residents only) 
10:00 to 11:00 
P Gothen (Universidade do Porto) Representations of surface groups and Higgs bundles  I
Classical Hodge theory uses harmonic forms as preferred representatives of cohomology classes. A representation of the fundamental group of a Riemann surface gives rise to a corresponding flat bundle. A Theorem of Donaldson and Corlette shows how to find a harmonic metric in this bundle. A flat bundle corresponds to class in first nonabelian cohomology and the Theorem can be viewed as an analogue of the classical representation of de Rham cohomology classes by harmonic forms.

INI 1  
11:00 to 11:30  Morning coffee  
11:30 to 12:30  Moduli spaces and physics  INI 1  
12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00  Geometric Invariant Theory  III  INI 1  
15:00 to 15:30  Afternoon tea  
15:30 to 16:30 
N Nitsure (Tata Institute of Fundamental Research) Deformation Theory : an introduction to the algebraic approach  III
This series of three lectures aims to give an introduction to the algebraic approach to deformation theory, as developed by Grothendieck, Schlessinger, Illusie and Artin, illustrated with a few key examples.
The topics will include (i) infinitesimal theory over Artin local rings including Schlessinger's theorem, tangentobstruction theories and use of cotangent complex, (ii) algebraization via the Grothendieck existence theorem for formal schemes and the Artin approximation theorem, and (iii) construction of global moduli as quotient algebraic spaces.

INI 1  
16:45 to 17:45  Tutorials  INI 1  
18:45 to 19:30  Dinner at Wolfson Court (residents only) 
10:00 to 11:00  Geometric Invariant Theory  IV  
11:00 to 11:30  Morning coffee  served at INI  
11:30 to 12:30 
P Gothen (Universidade do Porto) Representations of surface groups and Higgs bundles  II
A Higgs bundle on a Riemann surface is a pair consisting of a holomorphic bundle and a holomorphic oneform, the Higgs field, with values in a certain associated vector bundle. A theorem of Hitchin and Simpson says that a stable Higgs bundle admits a metric satisfying Hitchin's equations. Together with the Theorem of Corlette and Donaldson, the HitchinKobayashi correspondence generalizes the classical Hodge decomposition of the first cohomology of the Riemann surface, providing a correspondence between isomorphism classes of Higgs bundles and representations of the fundamental group of the surface.


12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Introduction to derived categories and stability conditions  I
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of tstructures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and KontsevichSoibelman) and discuss examples. The relevant literature is accessible on the arxiv.


15:00 to 15:30  Afternoon tea  served at INI  
15:30 to 16:30 
Introduction to moduli of varieties  I: Generalities on moduli spaces of varieties
The yoga of moduli spaces. Main methods of construction. Compactifications and Minimal Model Program. Stable pairs (slc & ample K+B). Honest work vs "accidental" examples.


16:45 to 17:45  Tutorials  INI 1  
18:45 to 19:30  Dinner at Wolfson Court (residents only) 
10:00 to 11:00 
Introduction to stacks  I
Introduction to algebraic stacks as they are relevant to moduli theory.


11:00 to 11:30  Morning coffee  served at INI  
11:30 to 12:30 
P Gothen (Universidade do Porto) Representations of surface groups and Higgs bundles  III
In the final lecture, we give some applications of Higgs bundle theory to the study of the geometry and topology of character varieties for surface groups, via the identification with moduli of Higgs bundles.


12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Introduction to derived categories and stability conditions  II
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of tstructures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and KontsevichSoibelman) and discuss examples. The relevant literature is accessible on the arxiv.


15:00 to 15:30  Afternoon tea  served at INI  
15:30 to 16:30 
Introduction to moduli of varieties  II: Stable toric and semiabelic varieties
Moduli of stable toric varieties vs toric Hilbert scheme. Convex tilings and secondary polytopes. Degenerations of abelian varieties. Semiabelian and semiabelic varieties. Compactifications of moduli spaces of abelian varieties.
Refs: http://arxiv.org/abs/math/9905103,
http://arxiv.org/abs/math/0207272, http://arxiv.org/abs/math/0207274.


16:45 to 17:45  Tutorials  INI 1  
18:45 to 19:30  Dinner at Wolfson Court (residents only) 
10:00 to 11:00 
Introduction to stacks  II
Introduction to algebraic stacks as they are relevant to moduli theory.


11:00 to 11:30  Morning coffee  served at INI  
11:30 to 12:30 
RingelHall algebras and applications to moduli  I
Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry.
The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representationtheoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields.
We will develop several such identities and discuss more recent applications to wallcrossing formulae.
Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wallcrossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.


12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Introduction to derived categories and stability conditions  III
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of tstructures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and KontsevichSoibelman) and discuss examples. The relevant literature is accessible on the arxiv.


15:00 to 15:30  Afternoon tea  served at INI  
15:30 to 16:30 
Introduction to moduli of varieties  III: Weighted hyperplane arrangements
$\bar M_{0,n}$. Hassett's weighted stable curves. Compact moduli of weighted hyperplane arrangements.
Refs: http://arxiv.org/abs/math/0205009,
http://arxiv.org/abs/math/0501227, http://arxiv.org/abs/0806.0881


16:45 to 17:45  Tutorials  INI 1  
19:30 to 22:00  Conference dinner at Emmanuel College 
10:00 to 11:00 
Introduction to stacks  III
Introduction to algebraic stacks as they are relevant to moduli theory.


11:00 to 11:30  Morning coffee  served at INI  
11:30 to 12:30 
RingelHall algebras and applications to moduli  II
Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry.
The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representationtheoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields.
We will develop several such identities and discuss more recent applications to wallcrossing formulae.
Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wallcrossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.


12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Introduction to derived categories and stability conditions  IV
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of tstructures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and KontsevichSoibelman) and discuss examples. The relevant literature is accessible on the arxiv.


15:00 to 15:30  Afternoon tea  served at INI  
15:30 to 16:30 
Introduction to moduli of varieties  IV: Surfaces of general type related to abelian varieties and hyperplane arrangements
Compact moduli of surfaces of general type derived from (1) abelian varieties, (2) line arrangements. Campedelli, Burniat, Kulikov surfaces.
Refs: http://arxiv.org/abs/math/9905103, http://arxiv.org/abs/0901.4431


16:45 to 17:45  Tutorials  INI 1  
18:45 to 19:30  Dinner at Wolfson Court (residents only) 
10:00 to 11:00 
Introduction to stacks  IV
Introduction to algebraic stacks as they are relevant to moduli theory.


11:00 to 11:30  Morning coffee  served at INI  
11:30 to 12:30 
RingelHall algebras and applications to moduli  III
Moduli spaces of representations of quivers, parametrizing configurations of vector spaces and linear maps up to base change, provide a prototype for many moduli spaces of algebraic geometry.
The Hall algebra of a quiver, a convolution algebra of functions on stacks of its representations, can be used to obtain quantitative information on the moduli spaces: algebraic identities in the Hall algebra, proved by representationtheoretic techniques, yield identities for e.g. Betti numbers or numbers of points over finite fields.
We will develop several such identities and discuss more recent applications to wallcrossing formulae.
Notes: Several of the Hall algebra techniques which I would like to discuss are reviewed in the survey "Moduli of representations of quivers", arXiv:0802.2147. Although this paper was written for an audience of representation theorists, it might as well be helpful for the participants of the School on Moduli Spaces. The more recent applications to wallcrossing formulae are developed in "Poisson automorphisms and quiver moduli", arXiv:0802.2147.


12:30 to 13:30  Lunch at Wolfson Court  
14:00 to 15:00 
Introduction to derived categories and stability conditions  V
The title may be misleading, I will actually assume the audience to be familiar with the basic concepts of derived and triangulated categories. (I might however recall the notion of tstructures.) The emphasis will be on stability conditions. I plan to introduce stability conditions in the sense of Bridgeland (and KontsevichSoibelman) and discuss examples. The relevant literature is accessible on the arxiv.


15:00 to 15:30  Afternoon tea  served at INI  
15:30 to 16:30 
Introduction to moduli of varieties  V: Surfaces of general type: the general case
Several versions of the moduli functor. Problems, and some solutions.
Steps in the construction of the moduli space.
Ref: many older papers, http://arxiv.org/abs/0805.0576,
http://arxiv.org/abs/1008.0621 (by Kollár)


16:45 to 17:45  Tutorials  INI 1  
18:45 to 19:30  Dinner at Wolfson Court (residents only) 